Here is a way to look at it and probably the method commonly used.

Differentiating the right one implicitly gives us:

Using the triangle we see that:

And [1] simplifies to

Therefore,

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To find

let

When you make the subs and simplify it whittles down to

Then, to get back in terms of x just resub

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This one simplifies wonderfully. Let

Make the subs and you finally get:

What is theta?.

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To tell when to use what, just remember the formulas:

, then use

, then use

, then use

Can you see whey we use those when we do?. Because they result in the proper identities that get rid of the radical.